Which Pair Of Equations Represents Perpendicular Lines?A. \[$ Y = 2x + 7 \$\] \[$ Y = -2x - 2 \$\]B. \[$ Y = 2x + 7 \$\] \[$ Y = X + 7 \$\]C. \[$ Y = 2x - 7 \$\] \[$ Y = -x - 7 \$\]D. \[$

Feb 27, 2025 by ADMIN 200 views

Understanding Perpendicular Lines

Perpendicular lines are two lines that intersect at a 90-degree angle. In the context of linear equations, perpendicular lines have slopes that are negative reciprocals of each other. In this article, we will explore which pair of equations represents perpendicular lines.

The Concept of Slope

The slope of a line is a measure of how steep it is. It is calculated as the ratio of the vertical change (rise) to the horizontal change (run) between two points on the line. The slope of a line can be represented by the letter 'm' and is calculated using the formula:

m = (y2 - y1) / (x2 - x1)

Slope of Perpendicular Lines

As mentioned earlier, the slopes of perpendicular lines are negative reciprocals of each other. This means that if the slope of one line is 'm', the slope of the other line will be '-1/m'. For example, if the slope of one line is 2, the slope of the other line will be -1/2.

Analyzing the Options

Let's analyze the options given to determine which pair of equations represents perpendicular lines.

Option A

${$ y = 2x + 7 $}$ ${$ y = -2x - 2 $}$

The slope of the first line is 2, and the slope of the second line is -2. Since the slopes are negative reciprocals of each other, this pair of equations represents perpendicular lines.

Option B

${$ y = 2x + 7 $}$ ${$ y = x + 7 $}$

The slope of the first line is 2, and the slope of the second line is 1. Since the slopes are not negative reciprocals of each other, this pair of equations does not represent perpendicular lines.

Option C

${$ y = 2x - 7 $}$ ${$ y = -x - 7 $}$

The slope of the first line is 2, and the slope of the second line is -1. Since the slopes are not negative reciprocals of each other, this pair of equations does not represent perpendicular lines.

Option D

${$ y = 2x + 7 $}$ ${$ y = -2x + 7 $}$

The slope of the first line is 2, and the slope of the second line is -2. Since the slopes are negative reciprocals of each other, this pair of equations represents perpendicular lines.

Conclusion

In conclusion, the pair of equations that represents perpendicular lines is Option A: ${$ y = 2x + 7 $}$ ${$ y = -2x - 2 $}$. This is because the slopes of the two lines are negative reciprocals of each other, which is a characteristic of perpendicular lines.

Key Takeaways

Perpendicular lines are two lines that intersect at a 90-degree angle.
The slopes of perpendicular lines are negative reciprocals of each other.
To determine if two lines are perpendicular, we need to check if their slopes are negative reciprocals of each other.

Practice Problems

Which pair of equations represents perpendicular lines?

${$ y = 3x + 2 $}$ ${$ y = -3x - 2 $}$

${$ y = 3x + 2 $}$ ${$ y = x + 2 $}$

${$ y = 3x - 2 $}$ ${$ y = -x - 2 $}$

${$ y = 3x + 2 $}$ ${$ y = -3x + 2 $}$

Which pair of equations represents perpendicular lines?

${$ y = 4x + 1 $}$ ${$ y = -4x - 1 $}$

${$ y = 4x + 1 $}$ ${$ y = x + 1 $}$

${$ y = 4x - 1 $}$ ${$ y = -x - 1 $}$

${$ y = 4x + 1 $}$ ${$ y = -4x + 1 $}$

Answer Key

Option A: ${$ y = 3x + 2 $}$ ${$ y = -3x - 2 $}$
Option A: ${$ y = 4x + 1 $}$ ${$ y = -4x - 1 $}$
Perpendicular Lines Q&A ==========================

Frequently Asked Questions

Q: What are perpendicular lines?

A: Perpendicular lines are two lines that intersect at a 90-degree angle. They are also known as orthogonal lines.

Q: How do I determine if two lines are perpendicular?

A: To determine if two lines are perpendicular, you need to check if their slopes are negative reciprocals of each other. If the slopes are negative reciprocals, then the lines are perpendicular.

Q: What is the slope of a line?

A: The slope of a line is a measure of how steep it is. It is calculated as the ratio of the vertical change (rise) to the horizontal change (run) between two points on the line.

Q: How do I calculate the slope of a line?

A: To calculate the slope of a line, you need to use the formula:

m = (y2 - y1) / (x2 - x1)

where m is the slope, and (x1, y1) and (x2, y2) are two points on the line.

Q: What is the difference between perpendicular lines and parallel lines?

A: Perpendicular lines intersect at a 90-degree angle, while parallel lines never intersect. The slopes of parallel lines are equal, while the slopes of perpendicular lines are negative reciprocals of each other.

Q: Can two lines be both perpendicular and parallel?

A: No, two lines cannot be both perpendicular and parallel. If two lines are perpendicular, they intersect at a 90-degree angle, while if two lines are parallel, they never intersect.

Q: How do I graph perpendicular lines?

A: To graph perpendicular lines, you need to draw two lines that intersect at a 90-degree angle. You can use a graphing calculator or a coordinate plane to help you draw the lines.

Q: What are some real-world examples of perpendicular lines?

A: Some real-world examples of perpendicular lines include:

The lines on a grid paper
The lines on a coordinate plane
The lines on a map
The lines on a building's blueprint

Q: Can I use perpendicular lines in real-world applications?

A: Yes, perpendicular lines have many real-world applications, including:

Architecture: Perpendicular lines are used to design buildings and structures.
Engineering: Perpendicular lines are used to design bridges and other infrastructure.
Art: Perpendicular lines are used to create geometric shapes and patterns.
Science: Perpendicular lines are used to measure angles and distances.

Q: How do I use perpendicular lines in math problems?

A: To use perpendicular lines in math problems, you need to identify the slopes of the lines and check if they are negative reciprocals of each other. You can then use the slopes to solve the problem.

Q: Can I use perpendicular lines to solve word problems?

A: Yes, perpendicular lines can be used to solve word problems. For example, if you are given a problem that involves finding the distance between two points, you can use perpendicular lines to solve it.

Common Mistakes to Avoid

Assuming that two lines are perpendicular just because they intersect at a point.
Failing to check if the slopes of two lines are negative reciprocals of each other.
Using the wrong formula to calculate the slope of a line.

Tips and Tricks

Use a graphing calculator or a coordinate plane to help you draw perpendicular lines.
Check if the slopes of two lines are negative reciprocals of each other before assuming that they are perpendicular.
Use the formula m = (y2 - y1) / (x2 - x1) to calculate the slope of a line.